# SLnEquivariantMatrices -- Ancillary file to the paper "A construction of equivariant bundles on the space of symmetric forms"

## Description

In the paper "A construction of equivariant bundles on the space of symmetric forms" (https://arxiv.org), the authors construct stable vector bundles on the space $\PP(S^d\CC^{n+1})$ of symmetric forms of degree $d$ in $n + 1$ variables which are equivariant for the action of $SL_{n+1}(\CC)$ ,and admit an equivariant free resolution of length 2.

Take two integers $d \ge 1$ and $m \ge 2$ and a vector spave $V = \CC^{n+1}$. For $n=2$, we have

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus S^{md-2}V \oplus S^{md-4}V \oplus \dots$,

while for $n > 1$,

$S^dV \otimes S^{(m-1)d}V = S^{md}V \oplus V_{(md-2)\lambda_1+\lambda_2} \oplus V_{(md-4)\lambda_1+2\lambda_2} \oplus \dots$,

where $\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of the Lie group $SL_{n+1}(\bf C)$ and $V_{i\lambda_1+j\lambda_2}$ is the irreducible representation of highest weight $i\lambda_1+j\lambda_2$.

The projection of the tensor product onto the second summand induces a $SL_{2}(\CC)$-equivariant morphism

$\Phi: S^{md-2}V \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V \otimes O_{\PP(S^dV)}(1)$

or a $SL_{n+1}(\CC)$-equivariant morphism

$\Phi: V_{(md-2)\lambda_1 + \lambda_2} \otimes O_{\PP(S^dV)} \to S^{(m-1)d}V \otimes O_{\PP(S^dV)}(1)$

with constant co-rank 1, and thus gives an exact sequence of vector bundles on $\PP(S^dV)$:

$0 \to W_{2,d,m} \to S^{md-2}V \otimes \mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V \otimes \mathcal{O}_{\PP(S^dV)}(1) \to \mathcal{O}_{\PP(S^dV)}(m) \to 0$,

$0 \to W_{n,d,m} \to V_{(md-2)\lambda_1 + \lambda_2} \otimes \mathcal{O}_{\PP(S^dV)} \to S^{(m-1)d}V \otimes \mathcal{O}_{\PP(S^dV)}(1) \to \mathcal{O}_{\PP(S^dV)}(m) \to 0$.

The package allows to compute

(1) the decomposition into irreducible $SL_{n+1}(\bf C)$-representations of the tensor product of two symmetric powers $S^a\CC^{n+1}$ and $S^b\CC^{n+1}$;

(2) the matrix representing the morphism $\Phi$;

(3) the vector bundle $W_{n,d,m}$.

## Version

This documentation describes version 1.0 of SLnEquivariantMatrices.

## Source code

The source code from which this documentation is derived is in the file SLnEquivariantMatrices.m2.

## For the programmer

The object SLnEquivariantMatrices is .